The CGM framework comprises a set of honest, non-malicious parties \(N := \lbrace 1, \ldots , n\rbrace \)  and their corresponding datasets \(D_1,\ldots ,D_n\) .
Let \(\mathcal {D}\)  be the true data distribution s.t. each party \(i\)  may only be able to sample its dataset \(D_i\)  from a restricted subset of the support of \(\mathcal {D}\) .
Every party \(i\)  sends \(D_i\)  to a trusted mediator who trains a generative model (e.g., GAN, variational autoencoder, or flow-based model) on the pooled dataset \(\bigcup _{i\in N}D_i\)  to produce a distribution \(\mathcal {G}\)  from which the mediator is able to draw samples. Informally, \(\mathcal {G}\)  represents an approximation of \(\mathcal {D}\) . The mediator then generates a large
synthetic dataset \(G\)  s.t. each synthetic data point in \(G\)  is drawn i.i.d. from \(\mathcal {G}\) . The reward to each party \(i\)  will be a subset \(G_i\)  (of synthetic data points) of \(G\)  and is thus said to be freely replicable.Like digital goods, model or data reward can be replicated at no marginal cost and given to more parties [1]}.
In this paper, we use the following definitions from cooperative game theory [2]}: A coalition \(C\)  is a subset of parties (i.e., \(C \subseteq N\) ). The grand coalition is the set \(N\)  of all parties. A coalition structure \(CS\)  is a partition of the parties into disjoint coalitions s.t. \(\bigcup _{C \in CS} C = N\) , \(C \cap C^{\prime } = \emptyset \)  for all \(C, C^{\prime } \in CS\)  and \(C \ne C^{\prime }\) ,
and each party cooperates only with parties in the same coalition.
A characteristic function \(v_c: 2^N \rightarrow \mathbb {R}\)  maps each coalition to a (real) value of the coalition.
Finally, the reward vector \((r_1, \ldots ,r_n) \in \mathbb {R}^{n}\)  denotes the final reward values assigned to parties \(1,\ldots ,n\) .
